Axisymmetric motion of two spherical particles in a Brinkman medium with slip surfaces

An effective-medium approach, based on the Brinkman equation is used to study the axisymmetric quasi-steady motion of two spherical particles embedded in a porous medium. The particles are in general of different sizes and translating with different velocities along the line connecting their centers, and allowing for the hydrodynamic slip at their surfaces. Under the Stokes flow approximation, a general solution is constructed using superposition of the basic solutions in two moving spherical coordinate systems based on the centers of the particles. A collocation technique is used to satisfy the boundary conditions on the surfaces of the particles. Numerical results for the normalized drag force acting on each particle are obtained with rapid convergence for various values of slip coefficients, size ratio, separation parameter, velocity ratio of the particles, and permeability parameter. The normalized drag force on each particle reach the single particle limit as the distance between centers grows large enough and each particle then may be translated independent of each other. The accuracy of the numerical technique has been tested against known solutions for two spheres with no-slip surfaces and when the porous medium becomes a clear fluid.